Verifying Uniqueness in a Logical Framework

Anderson, Penny; Pfenning, Frank

doi:10.1007/978-3-540-30142-4_2

Cited by 3 publications

(2 citation statements)

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“…Schürmann and Pfenning (1998) discuss a metatheorem prover. Anderson and Pfenning (2004) discuss uniqueness checking. describe an extended metatheorem language that would permit more general theorems than ∀∃-statements.…”

Section: Lf and Twelfmentioning

confidence: 98%

Mechanizing metatheory in a logical framework

Harper¹,

Licata²

2007

J. Funct. Prog.

107

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The LF logical framework codifies a methodology for representing deductive systems, such as programming languages and logics, within a dependently typed λ-calculus. In this methodology, the syntactic and deductive apparatus of a system is encoded as the canonical forms of associated LF types; an encoding is correct (adequate) if and only if it defines a compositional bijection between the apparatus of the deductive system and the associated canonical forms. Given an adequate encoding, one may establish metatheoretic properties of a deductive system by reasoning about the associated LF representation. The Twelf implementation of the LF logical framework is a convenient and powerful tool for putting this methodology into practice. Twelf supports both the representation of a deductive system and the mechanical verification of proofs of metatheorems about it.The purpose of this paper is to provide an up-to-date overview of the LF λ-calculus, the LF methodology for adequate representation, and the Twelf methodology for mechanizing metatheory. We begin by defining a variant of the original LF language, called Canonical LF, in which only canonical forms (long βη-normal forms) are permitted. This variant is parameterized by a subordination relation, which enables modular reasoning about LF representations. We then give an adequate representation of a simply typed λ-calculus in Canonical LF, both to illustrate adequacy and to serve as an object of analysis. Using this representation, we formalize and verify the proofs of some metatheoretic results, including preservation, determinacy, and strengthening. Each example illustrates a significant aspect of using LF and Twelf for formalized metatheory.

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Section: Lf and Twelfmentioning

confidence: 98%

Mechanizing metatheory in a logical framework

Harper¹,

Licata²

2007

J. Funct. Prog.

107

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show abstract

“…In order to realise this model, we shall use the machinery of logical frameworks (see, for example, Harper et al (1993) and Pfenning (2003), and, perhaps, Anderson and Pfenning (2004) and Harper and Pfenning (2005)). More precisely, we define our model of the Hybrid core subset as a theory in a logical framework.…”

Section: Modelling Hybrid In a Logical Frameworkmentioning

confidence: 99%

The representational adequacy of Hybrid

Crole

2011

Math. Struct. Comp. Sci.

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The Hybrid system (Ambler et al. 2002b), implemented within Isabelle/HOL, allows object logics to be represented using higher order abstract syntax (HOAS), and reasoned about using tactical theorem proving in general, and principles of (co)induction in particular. The form of HOAS provided by Hybrid is essentially a lambda calculus with constants.Of fundamental interest is the form of the lambda abstractions provided by Hybrid. The user has the convenience of writing lambda abstractions using names for the binding variables. However, each abstraction is actually a definition of a de Bruijn expression, andHybrid can unwind the user's abstractions (written with names) to machine friendly de Bruijn expressions (without names). In this sense the formal system contains a hybrid of named and nameless bound variable notation.In this paper, we present a formal theory in a logical framework, which can be viewed as a model of core Hybrid, and state and prove that the model is representationally adequate for HOAS. In particular, it is the canonical translation function from λ-expressions to Hybrid that witnesses adequacy. We also prove two results that characterise how Hybrid represents certain classes of λ-expression.We provide the first detailed proof to be published that proper locally nameless de Bruijn expressions and α-equivalence classes of λ-expressions are in bijective correspondence. This result is presented as a form of de Bruijn representational adequacy, and is a key component of the proof of Hybrid adequacy.The Hybrid system contains a number of different syntactic classes of expression, and associated abstraction mechanisms. Hence, this paper also aims to provide a self-contained theoretical introduction to both the syntax and key ideas of the system. Although this paper will be of considerable interest to those who wish to work with Hybrid in Isabelle/HOL, a background in automated theorem proving is not essential.

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An Executable Formalization of the HOL/Nuprl Connection in the Metalogical Framework Twelf

Schürmann

Stehr

2006

Logic for Programming, Artificial Intelligence, and Reasoning

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Verifying Uniqueness in a Logical Framework

Cited by 3 publications

References 20 publications

Mechanizing metatheory in a logical framework

Mechanizing metatheory in a logical framework

The representational adequacy of Hybrid

An Executable Formalization of the HOL/Nuprl Connection in the Metalogical Framework Twelf

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